For the proof, we refer to [1]}.
The expression in the first line naturally restricts to the partially symmetrized space \(\mathcal {H}_{\mathrm {r}}\otimes {\mathcal {H}}^\Sigma \) 
and the second expression to \({\mathcal {H}}^\Sigma \otimes \mathcal {H}_{\mathrm {r}}\) .
